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Minimum and maximum uncertainty values in quantum harmonic oscillator

  1. Jun 12, 2009 #1
    1. The problem statement, all variables and given/known data


    I have to find the minimum and maximum values of the uncertainty of [tex]\Delta[/tex]x and specify the times after t=0 when these uncertainties apply.

    2. Relevant equations

    The wave function is Ψ(x, 0) = (1/√2) (ψ1(x)+ iψ3(x))

    and for all t is Ψ(x, t) = (1/√2) (ψ1(x)exp(-3iwt/2+ iψ3(x)exp(-7iwt/2)

    3. The attempt at a solution

    the expectation value <x> = 0 ( given in my text )

    hence [tex]\Delta[/tex]x= [tex]\sqrt{}<x^2>[/tex]

    using the sandwich integral

    [tex]\int[/tex][tex]^{\infty}_{-\infty}[/tex]ψ1[tex]\ast[/tex](x) x^2 ψ1(x) dx = [tex]\frac{3}{2}[/tex] a^2

    [tex]\int[/tex][tex]^{\infty}_{-\infty}[/tex][tex]\psi[/tex]3[tex]\ast[/tex](x) x^2 [tex]\psi[/tex]3(x) dx = [tex]\frac{7}{2}[/tex]a^2

    [tex]\int[/tex][tex]^{\infty}_{-\infty}[/tex][tex]\psi[/tex]3[tex]\ast[/tex](x) x^2 [tex]\psi[/tex]1 (x) dx = [tex]\int[/tex][tex]^{\infty}_{-\infty}[/tex][tex]\psi[/tex][tex]\ast[/tex]1(x) x^2 [tex]\psi[/tex]3 dx = [tex]\sqrt{\frac{3}{2}}[/tex]a^2

    where a is the length parameter of the oscillator.


    Where do I go from here?
     
    Last edited: Jun 12, 2009
  2. jcsd
  3. Jun 12, 2009 #2

    Cyosis

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    You have calculated the probability density function correctly in a previous question you asked. Where has the time dependence disappeared? Secondly how does one find minimum/maximum values of a function generally?
     
  4. Jun 12, 2009 #3
    I don't know where the time dependence has gone

    I'm not sure how to calculate the minimum and maximum values of a function?
     
  5. Jun 12, 2009 #4
    I've been going at this for so long now my mind is turning mushy
     
  6. Jun 12, 2009 #5

    Cyosis

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    If you're following a course on quantum mechanics I'm pretty sure you know how to find the minimum/maximum of a function.

    Hint: derivative
     
  7. Jun 12, 2009 #6
    so i have to do a first derivative test
     
  8. Jun 12, 2009 #7

    Cyosis

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    First you need to find the correct time dependent expression for <x^2>. After that you take the derivative with respect to the appropriate variable and solve for said variable.
     
  9. Jun 12, 2009 #8
    I don't know how to find the time dependent expression for <x^2>
     
  10. Jun 12, 2009 #9

    Cyosis

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    In your other thread on this topic you calculated the probability density function for this time dependent wave function. The one with a sine in it and a t etc? You must remember.
    This is the correct "sandwich" expression to use. What you've calculated now is the time independent expectation value of x^2.

     
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